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Chapter 4 Differentiation in the Study of Calculus of Functions of One Variable, the Notions of Continuity, Differentiability and Integrability Play a Central Role

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Chapter 4 Differentiation in the Study of Calculus of Functions of One Variable, the Notions of Continuity, Differentiability and Integrability Play a Central Role Chapter 4 Differentiation In the study of calculus of functions of one variable, the notions of continuity, differentiability and integrability play a central role. The previous chapter was devoted to continuity and its consequences and the next chapter will focus on integrability. In this chapter we will define the derivative of a function of one variable and discuss several important consequences of differentiability. For example, we will show that differentiability implies continuity. We will use the definition of derivative to derive a few differentiation formulas but we assume the formulas for differentiating the most common elementary functions are known from a previous course. Similarly, we assume that the rules for differentiating are already known although the chain rule and some of its corollaries are proved in the solved problems. We shall not emphasize the various geometrical and physical applications of the derivative but will concentrate instead on the mathematical aspects of differentiation. In particular, we present several forms of the mean value theorem for derivatives, including the Cauchy mean value theorem which leads to L’Hôpital’s rule. This latter result is useful in evaluating so called indeterminate limits of various kinds. Finally we will discuss the representation of a function by Taylor polynomials. The Derivative Let fx denote a real valued function with domain D containing an L ? neighborhood of a point x0 5 D; i.e. x0 is an interior point of D since there is an L ; 0 such that NLx0 D. Then for any h such that 0 9 |h| 9 L, we can define the difference quotient for f near x0, fx + h ? fx D fx : 0 0 4.1 h 0 h It is well known from elementary calculus (and easy to see from a sketch of the graph of f near x0 ) that Dhfx0 represents the slope of a secant line through the points x0,fx0 and x0 + h,fx0 + h. Then we have Definition (Derivative) The function fx is said to be differentiable at the interior point r c 5 D if lim Dhfc exists. We denote the value of this limit by f c and refer to this as h0 the derivative of fx at x : c. We may also define the derivative of fx at the endpoints of an interval by using one sided limits for Dhfx. The set of points in D where the limit exists is the domain of a new function dy f rx, called the derivative of fx. We shall also use the notation for the derivative of dx fx. This derivative may be variously interpreted as: the slope of the tangent line to the graph of y : fx at the point, x,fx the instantaneous rate of change of y : fx with respect to x. It is important to note that while a secant line is a line through two points whose slope is then well defined, a tangent line is a line characterized by just one point, the point of tangency to the graph. Then the slope of the tangent line is not well defined but can only be defined in terms of a limit procedure. Similarly, the average speed of an object over a given time interval is well defined (it is the distance travelled during the time interval divided by the length of the time interval). Instantaneous speed at a given instant is not well defined but, 1 like the slope of the tangent line to a curve, can only be defined by a limit procedure. Definition (Derivatives of higher order) If the function f rx is differentiable, then its d2y derivative is denoted by f" x or . This is called the second derivative of dx 2 dny y : fx. We will denote derivatives of order higher than 2 by fnx or for n 5 N. dx n Example Derivatives (a) In an elementary calculus course, we derive formulas for the derivatives of many elementary functions. The following functions are differentiable at each point where they are defined: fx : xp f rx : px p?1 fx : Sin x f rx : Cos x fx : Cos x f rx : ?Sin x fx : ax, a ; 0 f rx : ax ln a : r : 1 fx ln x f x x (b)The following functions are continuous for all x, but the derivative fails to exist at the indicated points: fx : |x| is not differentiable at x : 0 since lim Dhf0 fails to exist (see problem 4.4 ) h0 fx : x s not differentiable at x : 0 since lim Dhf0 tends to +K as h 0 h0 The examples in (b) show that there are continuous functions that are not differentiable. The following theorem shows that there are no differentiable functions that fail to be continuous. Theorem 4.1 If fx is differentiable at x : c in D, then fx is continuous at x : c. Rules for Differentiation In elementary calculus we learn differentiation formulas for commonly occurring functions like the ones in example 4.1(a). In addition, we learn differentiation rules which allow us to compute derivatives of various combinations of functions when the derivatives of the separate functions are known. Theorem 4.2 (Derivatives of sums, products, quotients) Suppose fx and gx are differentiable at x : c in their common domain, D. Then 1) d C f + C gc : C f rc + C g rc for all constants C ,C dx 1 2 1 2 1 2 2) d f 6 gc : f rc 6 gc + fc 6 g rc 4.2 dx f f rc 6 gc ? fc 6 g rc 3) d c : if gc é 0. dx g gc2 Theorem 4.3 (Chain Rule) Suppose fx is differentiable at x : c and gx is differentiable at 2 y : fc. Then the composed function, Fx : gfx is differentiable at x : c and F rc : g rfc 6 f rc. Corollary 4.4 (Derivative of the inverse) Suppose f is strictly monotone and continuous on interval I. Then the inverse function g : f?1 is strictly monotone and continuous on the interval J : fI. Moreover, if f is differentiable at x : c in I and f rc é 0, then g is differentiable at d : fc and grd : 1 . f rc Corollary 4.5 (Derivative of parametric equations) Suppose x : ft and y : gt are differentiable functions of t for a t b, and that f rt é 0 for a t b. Then r dy : dy /dt : g t dx dx /dt f rt Consequences of Differentiability Just as there were a number of useful consequences of the property of continuity, there are similar consequences associated with differentiability. Local Extreme Points A point c in the domain D of a function fx is said to be a local maximum for fx if for some J ; 0, we have fc fx for all x 5 NJc V D. If fc fx for all x 5 NJc V D, then c is said to be a local minimum for fx. We say that c 5 D is a local extreme point for fx if it is either a local maximum or a local minimum. Theorem 4.6 (Extreme Points) Suppose fx is defined and continuous on the interval I and suppose c 5 I is a local extreme point for fx. Then exactly one of the following assertions must hold: i) c is an endpoint of the interval I ii) c is an interior point of I and f rc : 0 iii) c is an interior point of I but f rc fails to exist In an elementary calculus course students are taught that in order to find the extreme points of a function it is just a matter of computing the function’s derivative and setting it equal to zero. This theorem asserts that the extreme points of a function need not always be located at a point where its derivative is zero.(see problem 4.27). Mean Value Theorem for Derivatives There are several versions of one result, all of which are usually referred to as the mean value theorem for derivatives. The simplest version is known as Rolle’s theorem. Corollaries 4.8 and 4.9 and theorem 4.10 are generalizations of Rolle’s theorem. Theorem 4.7 (Rolle’s theorem) Suppose fx is continuous on the closed interval, a,b, and f is differentiable on the open interval a,b. Suppose further that fa : fb : 0. Then there exists a point, c 5 a,b, where f rc : 0. Corollary 4.8 (Mean value theorem for derivatives) Suppose fx is continuous on the closed 3 interval, a,b, and f is differentiable on the open interval a,b. Then there exists a point, c 5 a,b, where fb ? fa : f rcb ? a Corollary 4.9 Suppose fx is continuous on the closed interval, a,b, and f is differentiable on the open interval a,b. Suppose further that f rx : 0 for all x 5 a,b. Then f is constant on a,b. Theorem 4.10 (Cauchy’s Mean Value theorem) Suppose fx and gx continuous on the closed interval, a,b, and are differentiable on the open interval a,b. Suppose further that f rx,g rx never vanish simultaneously and that ga é gb. Then there exists a point, c 5 a,b, where ? r fb fa : f c gb ? ga g rc Indeterminate Forms A limit of a quotient in which the numerator and denominator tend to zero simultaneously is said to be an indeterminate limit of the form 0 . Often, such limits can be evaluated by 0 means of a corollary of the Cauchy mean value theorem known as L’Hôpital ’s rule .
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